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a+b=-12 ab=4\left(-7\right)=-28
Factor the expression by grouping. First, the expression needs to be rewritten as 4x^{2}+ax+bx-7. To find a and b, set up a system to be solved.
1,-28 2,-14 4,-7
Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -28.
1-28=-27 2-14=-12 4-7=-3
Calculate the sum for each pair.
a=-14 b=2
The solution is the pair that gives sum -12.
\left(4x^{2}-14x\right)+\left(2x-7\right)
Rewrite 4x^{2}-12x-7 as \left(4x^{2}-14x\right)+\left(2x-7\right).
2x\left(2x-7\right)+2x-7
Factor out 2x in 4x^{2}-14x.
\left(2x-7\right)\left(2x+1\right)
Factor out common term 2x-7 by using distributive property.
4x^{2}-12x-7=0
Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.
x=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\times 4\left(-7\right)}}{2\times 4}
All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.
x=\frac{-\left(-12\right)±\sqrt{144-4\times 4\left(-7\right)}}{2\times 4}
Square -12.
x=\frac{-\left(-12\right)±\sqrt{144-16\left(-7\right)}}{2\times 4}
Multiply -4 times 4.
x=\frac{-\left(-12\right)±\sqrt{144+112}}{2\times 4}
Multiply -16 times -7.
x=\frac{-\left(-12\right)±\sqrt{256}}{2\times 4}
Add 144 to 112.
x=\frac{-\left(-12\right)±16}{2\times 4}
Take the square root of 256.
x=\frac{12±16}{2\times 4}
The opposite of -12 is 12.
x=\frac{12±16}{8}
Multiply 2 times 4.
x=\frac{28}{8}
Now solve the equation x=\frac{12±16}{8} when ± is plus. Add 12 to 16.
x=\frac{7}{2}
Reduce the fraction \frac{28}{8} to lowest terms by extracting and canceling out 4.
x=-\frac{4}{8}
Now solve the equation x=\frac{12±16}{8} when ± is minus. Subtract 16 from 12.
x=-\frac{1}{2}
Reduce the fraction \frac{-4}{8} to lowest terms by extracting and canceling out 4.
4x^{2}-12x-7=4\left(x-\frac{7}{2}\right)\left(x-\left(-\frac{1}{2}\right)\right)
Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{7}{2} for x_{1} and -\frac{1}{2} for x_{2}.
4x^{2}-12x-7=4\left(x-\frac{7}{2}\right)\left(x+\frac{1}{2}\right)
Simplify all the expressions of the form p-\left(-q\right) to p+q.
4x^{2}-12x-7=4\times \frac{2x-7}{2}\left(x+\frac{1}{2}\right)
Subtract \frac{7}{2} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.
4x^{2}-12x-7=4\times \frac{2x-7}{2}\times \frac{2x+1}{2}
Add \frac{1}{2} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
4x^{2}-12x-7=4\times \frac{\left(2x-7\right)\left(2x+1\right)}{2\times 2}
Multiply \frac{2x-7}{2} times \frac{2x+1}{2} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.
4x^{2}-12x-7=4\times \frac{\left(2x-7\right)\left(2x+1\right)}{4}
Multiply 2 times 2.
4x^{2}-12x-7=\left(2x-7\right)\left(2x+1\right)
Cancel out 4, the greatest common factor in 4 and 4.